354 XV NORMED ALGEBRAS AND SPECTRAL THEORY (15.6.2.4), we have \f(z)\2 £f(e)f(z*z) g/(<?)2. This proves that / is con- tinuous and that ||/|| ^f(e) (5.7.1). Since also \\e\\ = 1, we have/(e) ^ ||/||, and assertion (i) is proved. To prove (ii), observe that for y e A the linear form x\-+f(y*xy) on A is positive, because f(y*x*xy) =f((xy)*(xy)) ^ 0; by virtue of (i), the norm of this linear form isf(y*y). This completes the proof. PROBLEMS 1. Let A be an algebra with involution and let U be a representation of A on a Hilbert space H. In order that U should be irreducible, it is necessary and sufficient that the subalgebra B of J5?(H) consisting of the endomorphisms Fsuch that VU(s) = U(s)V for all s e A should be equal to C • 1H. (To show that the condition is necessary, observe that B is an'involutive closed subalgebra of ^(H), hence is a star algebra. If S is a hermitian operator belonging to B, consider the closed commutative sub- algebra C of B generated by S, which is separable and therefore isornorphic to ^c(X) for some compact metrizable space X. Show that C has no divisors of zero, other than 0, and conclude that X consists of a single point.) 2. Let A be a Banach algebra with unit element e, and let x^x* be a (not necessarily continuous) involution on A. (a) Let / be a linear form on A such that f(a2) ^ 0 for all self-adjoint elements a e A. Show that if also p(a) < 1, then/(0) is real and |/(<a)| ^f(e) (use Section 15.4, Problem 17). Show that, for each self-adjoint element a e A, f(a) is real, \f(a)\ ^ f(e}p(a) and f(a)2 <.f(e)f(a2). (Consider f((a + ge)2) where f e R.) Deduce that \f(x)\ <^f(e)p(x), in the notation of Section 15,4, Problem 18, and a fortiori that /is continuous on A. Furthermore, we have \f(y*xy)\ ^p(x)f(y*y). (b) Conversely, if/is a linear form on A such that |/(*)| ^f(e)p(x) for all x e A, show that for each self-adjoint element a e A we have/0?) real and \f(a)\ "^f(e)p(a) (consider f(a + i£e\ where £ e R). (c) Suppose that the involution x>-*x* is hermitian (Section 15.4, Problem 18). Show that the following conditions are equivalent, for a linear form/on A: (a) /is a positive linear form, (j8) For every self-adjoint element a e A, f(a) is real and \f(a)\ ^ f(e)p(a). (y) For every self-adjoint element a e A,/(a2) > 0. (8) For each x e A, \f(x) \ <>f(e)p(x). (Use (a) and (b). To show that (/3) implies (a), observe that if Sp(a) is contained in an interval [A, JLC] <= R and if/(<?) =£ 0, then/(e) > 0 and f(a) e [f(e)\,f(e)p,l) 3. Let A be a separable Banach algebra with unit element e, and let x*-+-x* be a (not necessarily continuous) involution on A. (a) Show that if A is hermitian, then for each x e A there exists a positive linear form/on A such that /(<?)= 1 and f(x*x)l/2 =/?(*)• (Consider the commutative Banach subalgebra C generated by x*x, and a commutative Banach subalgebra B => C such that SpB(;y) = SpA0>) for all y e C. For each character x of B and eachthere exists a self-adjoint element b, commuting with a, such that b2 = <? — a2